The inclusion of elasticity is necessary to analyse the aforementioned systems. It was observed experimentally for red blood cells [1]} and numerically for flexible chains of beads [2]}, [3]} that high flexibility leads to a change in orientation (and sometimes shape) of sedimenting objects, affecting their sedimentation speed. Independently, in the case of linear filaments, [4]} observed that including elasticity can change the behaviour qualitatively when looking at bundling vs. non-bundling flagella. Even for a fixed shape, a change in orientation can alter the sedimentation velocity by \(25\%\) , as shown by [5]} and in further analytical solutions for sedimenting tori [6]}, [7]}, [8]}. Furthermore, as noted by [9]} and [10]}, dynamical buckling can occur in similar settings, resulting in significant shape changes of the filaments. In bead-spring models, [3]} observed the existence of an elasticity threshold beyond which flexible loops undergo significant changes in sedimentation dynamics. While stiff loops were seen to attain almost planar, oval shapes and sediment vertically or at an acute angle to gravity, depending on their stiffness, more flexible fibres exhibited a complex shape evolution. Our work aims to explore this stability threshold in slender-body dynamics, both analytically and in terms of numerical simulations.
